Summary. In this paper we will establish some necessary and/or sufficient conditions for both a nonsingular and a singular matrix A (interpreted as a linear map) to be pseudomonotone. The given results are in terms of the sign of the determinants of the principal submatrices and of the cofactors of A in the non singular case and in terms of the structure of A in the singular case. A complete characterization of pseudomonotonicity in terms of the coefficients of a Matrix of order 3 is given and a method for constructing a merely pseudomonotone matrix is suggested.